To find the equation of the line that is parallel to the given line and passes through a specific point, we can follow these steps:
1. Identify the slope of the given line: The given line is \(5x + 2y = 12\). We need to find its slope. To do this, we convert it to slope-intercept form \(y = mx + b\) where \(m\) is the slope.
Let's rewrite the line equation \(5x + 2y = 12\) in slope-intercept form:
[tex]\[
2y = -5x + 12
\][/tex]
[tex]\[
y = -\frac{5}{2}x + 6
\][/tex]
From this, we see that the slope \(m\) of the given line is \(-\frac{5}{2}\).
2. Determine the slope of the parallel line: Since parallel lines have the same slope, the slope of our new line will also be \(-\frac{5}{2}\).
3. Use the point-slope form of the line equation: We know the slope and a point \((-2, 4)\) through which the line passes. The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting \(m = -\frac{5}{2}\), \(x_1 = -2\), and \(y_1 = 4\) into the point-slope form:
[tex]\[
y - 4 = -\frac{5}{2}(x - (-2))
\][/tex]
Simplify:
[tex]\[
y - 4 = -\frac{5}{2}(x + 2)
\][/tex]
[tex]\[
y - 4 = -\frac{5}{2}x - 5
\][/tex]
[tex]\[
y = -\frac{5}{2}x - 1
\][/tex]
So, the equation of the line that is parallel to \(5x + 2y = 12\) and passes through the point \((-2, 4)\) is:
[tex]\[
y = -\frac{5}{2}x - 1
\][/tex]
Hence, the correct answer is:
[tex]\[y=-\frac{5}{2} x-1\][/tex]
